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We discuss some interesting sublattices of interval-valued fuzzy subgroups. In our main result, we consider the set of all interval-valued fuzzy normal subgroups with finite range that attain the same value at the identity element of the group. We then prove that this set forms a modular sublattice of the lattice of interval-valued fuzzy subgroups. In fact, this is an interval-valued fuzzy version of a well-known result from classical lattice theory. Finally, we employ a lattice diagram to exhibit the interrelationship among these sublattices.

In 1965, Zadeh [1] introduced the concept of a fuzzy set, and later generalized this to the notion of an interval-valued fuzzy set [2]. Since then, there has been tremendous interest in this subject because of the diverse range of applications, from engineering and computer science to social behavior studies. In particular, Gorzalczany [3] developed an inference method using interval-valued fuzzy sets.

In 1995, Biswas [4] studied interval-valued fuzzy subgroups. Subsequently, a number of researchers applied interval-valued fuzzy sets to algebra [5-11], and Lee et al. [12] furthered the investigation of interval-valued fuzzy subgroups in the sense of a lattice.

In this paper, we discuss some interesting sublattices of the lattice of interval-valued fuzzy subgroups of a group.

In the main result of our paper, we consider the set of all interval-valued fuzzy normal subgroups with finite range that attain the same value at the identity element of the group. We prove that this set forms a modular sublattice of the lattice of interval-valued fuzzy subgroups. In fact, this is an interval-valued fuzzy version of a well-known result from classical lattice theory. Finally, we use a lattice diagram to exhibit the interrelationship among these sublattices.

2. Preliminaries

In this section, we list some basic concepts and well-known results which are needed in the later sections. Throughout this paper, we will denote the unit interval [0, 1] as I. For any ordinary subset A on a set X, we will denote the characteristic function of A as χA.

Let D(I) be the set of all closed subintervals of the unit interval [0, 1]. The elements of D(I) are generally denoted by capital letters M,N, ···, and note that M = [ML,MU], where ML and MU are the lower and the upper end points respectively. Especially, we denote 0 = [0, 0], 1 = [1, 1], and a = [a, a] for every a

D(I)X. Then A is called an interval-valued fuzzy subgroupoid (IVGP) in X if AL(xy) ≥ AL(x) ∧ AL(y) and AU(xy) ≥ AU(x) ∧ AU(y), ∀x, yX. It is clear that

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,

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IVGP(X).

Definition 2.4[4]. Let A be an IVFs in a group G. Then A is called an interval-valued fuzzy subgroup (IVG) in G if it satisfies the conditions : For any x, y ∈G,

(i)AL(xy) ≥AL(x) ∧AL(y) andAU(xy) ≥AU(x) ∧AU(y).

(ii)AL(x−1) ≥AL(x) andAU(x−1) ≥AU(x).

We will denote the set of all IVGs of G as IVG(G).

Result 2.A[8, Proposition 4.3]. Let G be a group and let {Aα} α∈Γ ⊂ IVG(G). Then

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Aα ∈ IVG(G).

Result 2.B [4, Proposition 3.1]. Let A be an IVG in a group G. Then

(a)A(x−1) =A(x), ∀x∈G.

(b)AL(e) ≥AL(x) andAU(e) ≥AU(x), ∀x∈G, whereeis the identity ofG.

Result 2.C [8, Proposition 4.2]. Let G be a group and let A ⊂ G. Then A is a subgroup of G if and only if [χ A, χ A] ∈ IVG(G).

Definition 2.5[8]. Let A be an IVFS in a set X and let λ, μ ∈ I with λ ≤ μ. Then the set A [λ,μ] = {x ∈ X : AL(x) ≥ λ and AU(x) ≥ μ} is called a [λ, μ]-level subset of A.

3. Lattices of Interval-Valued Fuzzy Subgroups

In this section, we study the lattice structure of the set of intervalvalued fuzzy subgroups of a given group. From Definitions 2.1 and 2.2, we can see that for a set X, D(I)X forms a complete lattice under the usual ordering of interval-valued fuzzy inclusion ⊂, where the inf and the sup are the intersection and the union of interval-valued fuzzy sets, respectively. To construct the lattice of interval-valued fuzzy subgroups, we define the inf of a family Aα of interval-valued fuzzy subgroups to be the intersection ⋂Aα. However, the sup is defined as the interval-valued fuzzy subgroup generated by the union ⋃ Aα and denoted by ( ⋃ Aα). Thus we have the following result.

Proposition 3.1. Let G be a group. Then IVG(G) forms a complete lattice under the usual ordering of interval-valued fuzzy set inclusion ⊂.

Proof. Let {Aα}α

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be any subset of IVG(G). Then, by Result 2.A,

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∈ IVG(G). Moreover, it is clear that

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Aα is the largest interval-valued fuzzy subgroup contained in Aα for each

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. So

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Aα =

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Aα. On the other hand, we can easily see that (

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Aα) is the least intervalvalued fuzzy subgroup containing Aα for each

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. So

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Aα = (

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Aα). Hence IVG(G) is a complete lattice.

Next we construct certain sublattice of the lattice IVG(G). In fact, these sublattices reflect certain peculiarities of the intervalvalued fuzzy setting. For a group G, let IVGf (G) = {A ∈ IVG(G) : Im A is finite } and let IVG[s, t](G) = {A ∈ IVG(G) : A(e) = [s, t]}, where e is the identity of G. Then it is clear that IVGf (G)[resp. IVG[s, t](G)] is a sublattice of IVG(G). Moreover, IVGf (G)∩ IVG[s, t](G) is also a sublattice of IVG(G).

Definition 3.2[11]. Let (X, ·) be a groupoid and let A,B ∈ D(I)X. Then the interval-valued fuzzy product of A and B, denoted by A

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B, is an IVFS in X defined as follows : For each x ∈ X, Now to obtain our main results, we start with following two lemmas.

Proposition 3.5. Let G be a group and let A,B ∈ IVG(G). If Im A and Im B are finite, then for each [λ, μ] ∈ D(I), (AB) [λ, μ] = A [λ, μ] · B [λ, μ].

Definition 3.6[8]. Let G be a group and let A ∈ IVG(G). Then A is called interval-valued fuzzy normal subgroup (IVNG) of G if A(xy) = A(yx) for any x, y ∈ G.

We will denote the set of all IVNGs of G as IFNG(G). It is clear that if G is abelian, then every IVG of G is an IVNG of G.

Result 3.A [6, Proposition 2.13]. Let G be a group, let A ∈ IFNG(G) and let

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such that λ ≤ AL(e) and μ ≤ AU(e). Then A [λ, μ] ◁ G, where A [λ, μ] ◁ G means that A [λ, μ] is a normal subgroup of G.

Result 3.B [6, Proposition 2.17]. Let G be a group and let A ∈ IVG(G). If A [λ, μ] ◁G for each [λ, μ] ∈ Im A, Then A ∈ IVNG(G).

The following is the immediate result of Results 3.A and 3.B.

Theorem 3.7. Let G be a group and let A ∈ IVG(G). Then A ∈ IVNG(G) if and only if for each [λ, μ] ∈ Im A, A [λ, μ] ◁ G.

Result 3.C[8, Proposition 5.3]. Let G be a group and let A ∈ IVNG(G). If B ∈ IVG(G), then B

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A ∈IVG(G).

The following is the immediate result of Result 2.A and Definition 3.6.

Proposition 3.8. Let G be a group and let A, B ∈ IVNG(G). Then A ∩ B ∈ IVNG(G).

It is well-known that the set of all normal subgroups of a group forms a sublattice of the lattice of its subgroups. As an interval-valued fuzzy analog of this classical result we obtain the following result.

Theorem 3.9. Let G be a group and let IVNf[s, t](G) = {A ∈ IVNG(G) : Im A is finite and A(e) = [s, t]}. Then IVNf[s, t](G) is a sublattice of IVGf (G)∩ IVG[s, t](G). Hence IVNf[s, t](G) is a sublattice of IVG(G).

Let C ∈ IVG(G) such that A ⊂ C and B⊂ C. Let z ∈ G. Then Similarly, we have

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Thus A

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B ⊂ C. So A

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B = A ∨ B.

Now let [λ, μ] ∈ D(I). Since A,B ∈ IVNG(G), A [λ, μ]◁G and B [λ, μ]◁G. Then A (λ,μ)

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B [λ, μ]◁G. By Proposition 3.5, (A

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B) [λ, μ] ◁ G. Thus, by Theorem 3.7, A

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B ∈ IVNG(G). So A ∨ B ∈ IVNf[s, t](G). From Proposition 3.8, it is clear that A ∧ B ∈ IVNG(G). Thus A ∧ B ∈ IVNf[s,t](G). Hence IVNf[s,t](G) is a sublattice of IVGf∩ IVG[s,t](G), and therefore of IVG(G). This complete the proof.

The relationship of different sublattice of the lattice of interval-valued fuzzy subgroup discussed herein can be visualized by the lattice diagram in Figure 1.

It is also well-known[20, Theorem I.11] that the sublattice of normal subgroups of a group is modular. As an interval-valued fuzzy version to the classical theoretic result, we prove that IVN([s, t](G) forms a modular lattice.

Result 3.D [11, Lemma 3.2]. Let G be a group and let A ∈ IVG(G). If for any x, y ∈ G, A L(x) < A L(y) and A U(x) < A U(y), then A(xy) = A(x) = A(yx).

In this paper, we mainly study sublattices of the lattice of interval-valued fuzzy subgroups of a group. In particular, we prove that the lattice IVNf[s, t](G) is modular lattice (See Theorem 3.11). Finally, for subgroup S(G) of a group G, IVG(S(G)) forms a sublattice of IVGf (G)∩ IVG[1,1](G) and hence of IVG(G) (See Proposition 3.14).

In the future, we will investigate sublattices of the lattice of interval-valued fuzzy subrings of a ring.

Conflict of InterestNo potential conflict of interest relevant to this article was reported.

Acknowledgements

This work was supported by the research grant of theWonkwang University in 2014.

BIO

Jeong Gon Lee received the Ph.D degree in The Department of Mathematics Education from Korea National University of Education. He is currently Assistant Professor inWonkwang University, Korea. His research interests are Measure Theory, Operator Theory, Mathematics Education, Category Theory, Hyperspace, and Topology. At present he has worked as one of ”Managing Editors” in Annals of Fuzzy Mathematics and Informatics (AFMI).

E-mail: jukolee@wku.ac.kr

Kul Hur received the Ph.D degree in The Department of Mathematics from Yonsei University. He was a Professor inWonkwang University. His research interests are Category Theory, Hyperspace and Topology. He retired from Wonkwang University on February 2012. At present he has worked as one of ”Editors-in-Chief” in Annals of Fuzzy Mathematics and Informatics (AFMI).

E-mail: kulhur@wonkwang.ac.kr

Pyung Ki Lim received the Ph.D degree in The Department of Mathematics from Chonnam National University, Korea. He is currently Professor in Wonkwang University. His research interests are Category Theory, Hyperspace and Topology.